Relations between generalised Gelfand-Tsetlin and Kazhdan-Lusztig bases of $S_n$

We prove that the Kazhdan-Lusztig basis of Specht modules is upper triangular with respect to all generalized Gelfand-Tsetlin bases constructed from any multiplicity-free tower of standard parabolic subgroups.
Comment: 10 pages, extended abstract for submission to FPSAC 2025